Leibniz algebras in characteristic

Bipartite Graphs and the Structure of Finite-Dimensional Semisimple Leibniz Algebras

“…Left and right actions induce Lie algebra structure on M 'g, where M becomes an abelian ideal and g is a subalgebra. If one has the right action of g on M and sets the left action to be zero, this induces a new type of a product that generalizes the Lie bracket on g given in the following A similar construction is given in [4].…”

“…In this paper a "non-commutative" generalization of Lie algebras, introduced by Bloh ( [3]) and later by Loday ([8], [9])-the so-called Leibniz algebras are studied. Although the classical simplicity for Leibniz algebras implies that it is a Lie algebra, a modified definition of the simplicity was introduced in [4] and has been in use in the various papers on the structure theory of Leibniz algebras. Generalization of semisimplicity for Leibniz algebras draws a parallel with semisimple Lie algebras, which is the main focus of the current work.…”

Description of some classes of Leibniz algebras

“…The definition of simplicity for Leibniz algebras has been suggested by Dzhumadil'daev in [3] as algebra L having the only ideals {0}, I and L. However, in order to eschew the solvability of L, the reasonable definition of the simplicity must be as follows. Obviously, in the case when the Leibniz algebra L is Lie, the ideal I is trivial and this definition agrees with the classical definition of simple Lie algebra.…”

“…The problem (2) for Leibniz algebras is still remaining untouched. This paper presents a progress made in the problem (3). It deals with the description of some classes of semisimple complex Leibniz algebras.…”

F[x,y] as a Dialgebra and a Leibniz Algebra

“…[1][2][3][4][5][6][7][8][9][10][11][12]). It is well-known that any associative algebra gives rise to a Lie algebra by x y = xy − yx.…”